A summation with Stirling numbers of the Second kind

Question

Can the following sum be evaluated?

$$\sum\limits_{k=1}^{n-1}\binom{n}{n-k}\left( n-k\right) !S\left( n,n-k\right) \left( \frac{k-1}{k}\right) ^{k}$$

where $S(n,m)$ is the Stirling number of the second kind.

Answer 1

Hint:

use :

$$n^n=\sum_{k=1}^{n}\binom{n}{n-k}\left(n-k\right)!{n\brace n-k}$$ And observe that ${n\brace 0}=0$ for $n>0$